By Fatskills Exam Guides Team — the exam nerds behind 28,500+ quizzes and 2.1M practice questions across 500+ global exams.
Units of Measurement Metric Measurement Prefixes Giga-: one billion (1 gigawatt is one billion watts) Mega-: one million (1 megahertz is one million hertz) Kilo-: one thousand (1 kilogram is one thousand grams) Deci-: one tenth (1 decimeter is one tenth of a meter) Centi-: one hundredth (1 centimeter is one hundredth of a meter) Milli-: one thousandth (1 milliliter is one thousandth of a liter) Micro-: one millionth (1 microgram is one millionth of a gram) Measurement Conversion When converting between units, the goal is to maintain the same meaning but change the way it is displayed. In order to go from a larger unit to a smaller unit, multiply the number of the known amount by the equivalent amount. When going from a smaller unit to a larger unit, divide the number of the known amount by the equivalent amount. For complicated conversions, it may be helpful to set up conversion fractions. In these fractions, one fraction is the conversion factor. The other fraction has the unknown amount in the numerator. So, the known value is placed in the denominator. Sometimes the second fraction has the known value from the problem in the numerator, and the unknown in the denominator. Multiply the two fractions to get the converted measurement. Note that since the numerator and the denominator of the factor are equivalent, the value of the fraction is 1. That is why we can say that the result in the new units is equal to the result in the old units even though they have different numbers. It can often be necessary to chain known conversion factors together. As an example, consider converting 512 square inches to square meters. We know that there are 2.54 centimeters in an inch and 100 centimeters in a meter, and that we will need to square each of these factors to achieve the conversion we are looking for. Common Units and Equivalents Metric Equivalents 1000 μg (microgram) - 1 mg 1000 mg (milligram) - 1 g 1000 g (gram) - 1 kg 1000 kg (kilogram) - 1 metric ton 1000 mL (milliliter) - 1 L 1000 μm (micrometer) - 1 mm 1000 mm (millimeter) - 1 m 100 cm (centimeter) - 1 m 1000 m (meter) - 1 km Distance and Area Measurement Unit - Abbreviation - U.S. equivalent - Metric equivalent Inch - in - 1 inch - 2.54 centimeters Foot - ft - 12 inches - 0.305 meters Yard - yd - 3 feet - 0.914 meters Mile - mi - 5280 feet - 1.609 kilometers Acre - ac - 4840 square yards - 0.405 hectares Square Mile 640 acres - 2.590 square kilometers Capacity Measurements Unit - Abbreviation - U.S. equivalent - Metric equivalent Fluid Ounce - fl oz - 8 fluid drams - 29.573 milliliters Cup - cp - 8 fluid ounces - 0.237 liter Pint - pt - 16 fluid ounces - 0.473 liter Quart - qt - 2 pints - 0.946 liter Gallon - gal - 4 quarts - 3.785 liters Teaspoon - t or tsp - 1 fluid dram - 5 milliliters Tablespoon - T or tbsp - 4 fluid drams - 15 or 16 milliliters Cubic Centimeter - cc or - 0.271 drams - 1 milliliter Weight Measurements Unit - Abbreviation - U.S. equivalent - Metric equivalent Ounce - oz - 16 drams - 28.35 grams Pound - lb - 16 ounces - 453.6 grams Ton - t - 2,000 pounds - 907.2 kilograms Volume and Weight Measurement Clarifications Always be careful when using ounces and fluid ounces. They are not equivalent. Having one pint of something does not mean you have one pound of it. In the same way, just because something weighs one pound does not mean that its volume is one pint. In the United States, the word 'ton' by itself refers to a short ton or a net ton. Do not confuse this with a long ton (also called a gross ton) or a metric ton (also spelled tonne), which have different measurement equivalents. Rounding and Estimation Rounding is reducing the digits in a number while still trying to keep the value similar. The result will be less accurate, but will be in a simpler form, and will be easier to use. Whole numbers can be rounded to the nearest ten, hundred or thousand. When you are asked to estimate the solution to a problem, you will need to provide only an approximate figure or estimation for your answer. In this situation, you will need to round each number in the calculation to the level indicated (nearest hundred, nearest thousand, etc.) or to a level that makes sense for the numbers involved. When estimating a sum all numbers must be rounded to the same level. You cannot round one number to the nearest thousand while rounding another to the nearest hundred. P1. Perform the following conversions: (a) 1.4 meters to centimeters (b) 218 centimeters to meters (c) 42 inches to feet (d) 15 kilograms to pounds (e) 80 ounces to pounds (f) 2 miles to kilometers (g) 5 feet to centimeters (h) 15.14 liters to gallons (i) 8 quarts to liters (j) 13.2 pounds to grams P2. Round each number to the indicated degree: Round to the nearest ten: 11; 47; 118 Round to the nearest hundred: 78; 980; 248 (c) Round each number to the nearest thousand: 302; 1274; 3756 P3. Estimate the solution to by rounding each number to the nearest ten thousand. P4. A runner's heart beats 422 times over the course of six minutes. About how many times did the runner's heart beat during each minute? P1. (a) Cross multiply to get (b) Cross multiply to get , or (c) Cross multiply to get , or (d) (e) (f) (g) (h) (i) (j) P2. (a) When rounding to the nearest ten, anything ending in 5 or greater rounds up. So, 11 rounds to 10, 47 rounds to 50, and 118 rounds to 120. (b) When rounding to the nearest hundred, anything ending in 50 or greater rounds up. So, 78 rounds to 100, 980 rounds to 1000, and 248 rounds to 200. (c) When rounding to the nearest thousand, anything ending in 500 or greater rounds up. So, 302 rounds to 0, 1274 rounds to 1000, and 3756 rounds to 4000. P3. Start by rounding each number to the nearest ten thousand: 345,932 becomes 350,000, and 96,369 becomes 100,000. Then, add the rounded numbers: . So, the answer is approximately 450,000. The exact answer would be . So, the estimate of 450,000 is a similar value to the exact answer. P4. 'About how many' indicates that you need to estimate the solution. In this case, look at the numbers you are given. 422 can be rounded down to 420, which is easily divisible by 6. A good estimate is beats per minute. More accurately, the patient's heart rate was just over 70 beats per minute since his heart actually beat a little more than 420 times in six minutes. Factoring Factors and Greatest Common Factor Factors are numbers that are multiplied together to obtain a product. For example, in the equation , the numbers 2 and 3 are factors. A prime number has only two factors (1 and itself), but other numbers can have many factors. A common factor is a number that divides exactly into two or more other numbers. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12, while the factors of 15 are 1, 3, 5, and 15. The common factors of 12 and 15 are 1 and 3. A prime factor is also a prime number. Therefore, the prime factors of 12 are 2 and 3. For 15, the prime factors are 3 and 5. The greatest common factor (GCF) is the largest number that is a factor of two or more numbers. For example, the factors of 15 are 1, 3, 5, and 15; the factors of 35 are 1, 5, 7, and 35. Therefore, the greatest common factor of 15 and 35 is 5. Multiples and Least Common Multiple Often listed out in multiplication tables, multiples are integer increments of a given factor. In other words, dividing a multiple by the factor number will result in an integer. For example, the multiples of 7 include: . Dividing 7, 14, 21, 28, or 35 by 7 will result in the integers 1, 2, 3, 4, and 5, respectively. The least common multiple (LCM) is the smallest number that is a multiple of two or more numbers. For example, the multiples of 3 include 3, 6, 9, 12, 15, etc.; the multiples of 5 include 5, 10, 15, 20, etc. Therefore, the least common multiple of 3 and 5 is 15. Fractions A fraction is a number that is expressed as one integer written above another integer, with a dividing line between them x divided by y.' It can also be thought of as x out of y equal parts. The top number of a fraction is called the numerator, and it represents the number of parts under consideration. The 1 in means that 1 part out of the whole is being considered in the calculation. The bottom number of a fraction is called the denominator, and it represents the total number of equal parts. The 4 in means that the whole consists of 4 equal parts. A fraction cannot have a denominator of zero; this is referred to as 'undefined.' Fractions can be manipulated, without changing the value of the fraction, by multiplying or dividing (but not adding or subtracting) both the numerator and denominator by the same number. If you divide both numbers by a common factor, you are reducing or simplifying the fraction. Two fractions that have the same value but are expressed differently are known as equivalent fractions. For example, are all equivalent fractions. They can also all be reduced or simplified to . When two fractions are manipulated so that they have the same denominator, this is known as finding a common denominator. The number chosen to be that common denominator should be the least common multiple of the two original denominators. Example: the least common multiple of 4 and 6 is 12. Manipulating to achieve the common denominator: . Proper Fractions and Mixed Numbers A fraction whose denominator is greater than its numerator is known as a proper fraction, while a fraction whose numerator is greater than its denominator is known as an improper fraction. Proper fractions have values less than one and improper fractions have values greater than one. A mixed number is a number that contains both an integer and a fraction. Any improper fraction can be rewritten as a mixed number. Example: . Similarly, any mixed number can be rewritten as an improper fraction. Example: . Adding and Subtracting Fractions If two fractions have a common denominator, they can be added or subtracted simply by adding or subtracting the two numerators and retaining the same denominator. If the two fractions do not already have the same denominator, one or both of them must be manipulated to achieve a common denominator before they can be added or subtracted. Example: . Multiplying Fractions Two fractions can be multiplied by multiplying the two numerators to find the new numerator and the two denominators to find the new denominator. Example: . Dividing Fractions Two fractions can be divided by flipping the numerator and denominator of the second fraction and then proceeding as though it were a multiplication. Example: . Multiplying a Mixed Number by a Whole Number or a Decimal When multiplying a mixed number by something, it is usually best to convert it to an improper fraction first. Additionally, if the multiplicand is a decimal, it is most often simplest to convert it to a fraction. For instance, to multiply by 3.5, begin by rewriting each quantity as a whole number plus a proper fraction. Remember, a mixed number is a fraction added to a whole number and a decimal is a representation of the sum of fractions, specifically tenths, hundredths, thousandths, and so on: Next, the quantities being added need to be expressed with the same denominator. This is achieved by multiplying and dividing the whole number by the denominator of the fraction. Recall that a whole number is equivalent to that number divided by 1: When multiplying fractions, remember to multiply the numerators and denominators separately: Now that the fractions have the same denominators, they can be added: Finally, perform the last multiplication and then simplify: Decimals Decimals are one way to represent parts of a whole. Using the place value system, each digit to the right of a decimal point denotes the number of units of a corresponding negative power of ten. For example, consider the decimal 0.24. We can use a model to represent the decimal. Since a dime is worth one-tenth of a dollar and a penny is worth one-hundredth of a dollar, one possible model to represent this fraction is to have 2 dimes representing the 2 in the tenths place and 4 pennies representing the 4 in the hundredths place: To write the decimal as a fraction, put the decimal in the numerator with 1 in the denominator. Multiply the numerator and denominator by tens until there are no more decimal places. Then simplify the fraction to lowest terms. For example, converting 0.24 to a fraction: Adding and Subtracting Decimals When adding and subtracting decimals, the decimal points must always be aligned. Adding decimals is just like adding regular whole numbers. Example: . If the problem-solver does not properly align the decimal points, an incorrect answer of 4.7 may result. A. easy way to add decimals is to align all of the decimal points in a vertical column visually. This will allow one to see exactly where the decimal should be placed in the final answer. Begin adding from right to left. Add each column in turn, making sure to carry the number to the left if a column adds up to more than 9. The same rules apply to the subtraction of decimals. Multiplying Decimals A simple multiplication problem has two components: a multiplicand and a multiplier. When multiplying decimals, work as though the numbers were whole rather than decimals. Once the final product is calculated, count the number of places to the right of the decimal in both the multiplicand and the multiplier. Then, count that number of places from the right of the product and place the decimal in that position. For example, has a total of three places to the right of the respective decimals. Multiply to get 31488. Now, beginning on the right, count three places to the left and insert the decimal. The final product will be 31.488. Dividing Decimals Every division problem has a divisor and a dividend. The dividend is the number that is being divided. In the problem , 14 is the dividend and 7 is the divisor. In a division problem with decimals, the divisor must be converted into a whole number. Begin by moving the decimal in the divisor to the right until a whole number is created. Next, move the decimal in the dividend the same number of spaces to the right. For example, 4.9 into 24.5 would become 49 into 245. The decimal was moved one space to the right to create a whole number in the divisor, and then the same was done for the dividend. Once the whole numbers are created, the problem is carried out normally: Percentages Percentages can be thought of as fractions that are based on a whole of 100; that is, one whole is equal to 100%. The word percent means "per hundred." Percentage problems are often presented in three main ways: · Find what percentage of some number another number is. o Example: What percentage of 40 is 8? · Find what number is some percentage of a given number. number is 20% of 40? Find what number another number is a given percentage of. o Example: What number is 8 20% of? There are three components in each of these cases: a whole (W), a part (P), and a percentage (%). These are related by the equation: . This can easily be rearranged into other forms that may suit different questions better: and . Percentage problems are often also word problems. As such, a large part of solving them is figuring out which quantities are what. For example, consider the following word problem: In a school cafeteria, 7 students choose pizza, 9 choose hamburgers, and 4 choose tacos. What percentage of student choose tacos? To find the whole, you must first add all of the parts: . The percentage can then be found by dividing the part by the whole ): . Converting Between Percentages, Fractions, and Decimals Converting decimals to percentages and percentages to decimals is as simple as moving the decimal point. To convert from a decimal to a percentage, move the decimal point two places to the right. To convert from a percentage to a decimal, move it two places to the left. It may be helpful to remember that the percentage number will always be larger than the equivalent decimal number. For example: To convert a fraction to a decimal, simply divide the numerator by the denominator in the fraction. To convert a decimal to a fraction, put the decimal in the numerator with 1 in the denominator. Multiply the numerator and denominator by tens until there are no more decimal places. Then simplify the fraction to lowest terms. For example, converting 0.24 to a fraction: Fractions can be converted to a percentage by finding equivalent fractions with a denominator of 100. For example, To convert a percentage to a fraction, divide the percentage number by 100 and reduce the fraction to its simplest possible terms. For example, Rational Numbers The term rational means that the number can be expressed as a ratio or fraction. That is, a number, r, is rational if and only if it can be represented by a fraction . Numbers P1. What is 30% of 120? P2. What is 150% of 20? P3. What is 14.5% of 96? P4. Simplify the following expressions: (a) (b) (d) (e) P5. Convert the following to a fraction and to a decimal: (a) 15%; (b) 24.36% P6. Convert the following to a decimal and to a percentage. (a) 4/5; (b) P7. A woman's age is thirteen more than half of 60. How old is the woman? P8. A patient was given pain medicine at a dosage of 0.22 grams. The patient's dosage was then increased to 0.80 grams. By how much was the patient's dosage increased? P9. At a hotel, of the 100 rooms are occupied today. Yesterday, of the 100 rooms were occupied. On which day were more of the rooms occupied and by how much more? P10. At a school, 40% of the teachers teach English. If 20 teachers teach English, how many teachers work at the school? P11. A patient was given blood pressure medicine at a dosage of 2 grams. The patient's dosage was then decreased to 0.45 grams. By how much was the patient's dosage decreased? P12. Two weeks ago, of the 60 customers at a skate shop were male. Last week, of the 80 customers were male. During which week were there more male customers? P13. Jane ate lunch at a local restaurant. She ordered a $4.99 appetizer, a $12.50 entrée, and a $1.25 soda. If she wants to tip her server 20%, how much money will she spend in all? P14. According to a survey, about 82% of engineers were highly satisfied with their job. If 145 engineers were surveyed, how many reported that they were highly satisfied? P15. A patient was given 40 mg of a certain medicine. Later, the patient's dosage was increased to 45 mg. What was the percent increase in his medication? P16. Order the following rational numbers from least to greatest: 0.55, 17%, , , , 3. P17. Order the following rational numbers from greatest to least: 0.3, 27%, , , . 4.5 P18. Perform the following multiplication. Write each answer as a mixed number. (a) (b) (c) P19. Suppose you are making doughnuts and you want to triple the recipe you have. If the following list is the original amounts for the ingredients, what would be the amounts for the tripled recipe? cup Flour tsp Baking powder Salt Sugar Tbsp Butter large Eggs Vanilla extract Sour cream P1. The word of indicates multiplication, so 30% of 120 is found by multiplying 120 by 30%. Change 30% to a decimal, then multiply: P2. The word of indicates multiplication, so 150% of 20 is found by multiplying 20 by 150%. Change 150% to a decimal, then multiply: P3. Change 14.5% to a decimal before multiplying. . P4. Follow the order of operations and utilize properties of fractions to solve each: Rewrite the problem as a multiplication problem: . Make sure the fraction is reduced to lowest terms. Both 14 and 20 can be divided by 2. The denominators of and are 8 and 16, respectively. The lowest common denominator of 8 and 16 is 16 because 16 is the least common multiple of 8 and 16. Convert the first fraction to its equivalent with the newly found common denominator of 16: . Now that the fractions have the same denominator, you can subtract them. When simplifying expressions, first perform operations within groups. Within the set of parentheses are multiplication and subtraction operations. Perform the multiplication first to get . Then, subtract two to obtain Finally, perform addition from left to right: (d) First, evaluate the terms in the parentheses using order of operations. , and . Next, rewrite the problem: . Finally, add and subtract from left to right: ; . The answer is . First, simplify within the parentheses, then change the fraction to a decimal and perform addition from left to right: P5. (a) 15% can be written as . Both 15 and 100 can be divided by 5: When converting from a percentage to a decimal, drop the percent sign and move the decimal point two places to the left: (b) 24.36% written as a fraction is , or , which reduces to . 24.36% written as a decimal is 0.2436. Recall that dividing by 100 moves the decimal two places to the left. P6. (a) Recall that in the decimal system the first decimal place is one tenth: Percent means 'per hundred.' (b) The mixed number has a whole number and a fractional part. The fractional part can be written as a decimal by dividing 5 into 2, which gives 0.4. Adding the whole to the part gives 3.4. To find the equivalent percentage, multiply the decimal by 100. . Notice that this percentage is greater than 100%. This makes sense because the original mixed number is greater than 1. P7. 'More than' indicates addition, and 'of' indicates multiplication. The expression can be written as . So, the woman's age is equal to . The woman is 43 years old. P8. The first step is to determine what operation (addition, subtraction, multiplication, or division) the problem requires. Notice the keywords and phrases 'by how much' and 'increased.' 'Increased' means that you go from a smaller amount to a larger amount. This change can be found by subtracting the smaller amount from the larger amount: grams. Remember to line up the decimal when subtracting:0.800.220.58 P9. First, find the number of rooms occupied each day. To do so, multiply the fraction of rooms occupied by the number of rooms available: The difference in the number of rooms occupied is: P10. To answer this problem, first think about the number of teachers that work at the school. Will it be more or less than the number of teachers who work in a specific department such as English? More teachers work at the school, so the number you find to answer this question will be greater than 20. 40% of the teachers are English teachers. 'Of' indicates multiplication, and words like 'is' and 'are' indicate equivalence. Translating the problem into a mathematical sentence gives , where t represents the total number of teachers. Solving for t gives . Fifty teachers work at the school. P11. The decrease is represented by the difference between the two amounts: Remember to line up the decimal point before subtracting.2.000.451.55 P12. First, you need to find the number of male customers that were in the skate shop each week. You are given this amount in terms of fractions. To find the actual number of male customers, multiply the fraction of male customers by the number of customers in the store.
The number of male customers was the same both weeks. P13. To find total amount, first find the sum of the items she ordered from the menu and then add 20% of this sum to the total. P14. 82% of 145 is . Because you can't have 0.9 of a person, we must round up to say that 119 engineers reported that they were highly satisfied with their jobs. P15. To find the percent increase, first compare the original and increased amounts. The original amount was 40 mg, and the increased amount is 45 mg, so the dosage of medication was increased by 5 mg ( ). Note, however, that the question asks not by how much the dosage increased but by what percentage it increased. P16. Recall that the term rational simply means that the number can be expressed as a ratio or fraction. Notice that each of the numbers in the problem can be written as a decimal or integer: So, the answer is 17%, , 0.55, 3, , . P17. Converting all the numbers to integers and decimals makes it easier to compare the values: So, the answer is , , 4.5, 0.3, 27%, . Rational Numbers P18. For each, convert improper fractions, adjust to a common denominator, perform the operations, and then simplify: Sometimes, you can skip converting the denominator and just distribute the multiplication. (b) (c) P19. Fortunately, some of the amounts are duplicated, so we do not need to figure out every amount. So, the result for the triple recipe is: Proportions A proportion is a relationship between two quantities that dictates how one changes when the other changes. A direct proportion describes a relationship in which a quantity increases by a set amount for every increase in the other quantity, or decreases by that same amount for every decrease in the other quantity. Example: Assuming a constant driving speed, the time required for a car trip increases as the distance of the trip increases. The distance to be traveled and the time required to travel are directly proportional. Inverse proportion is a relationship in which an increase in one quantity is accompanied by a decrease in the other, or vice versa. Example: the time required for a car trip decreases as the speed increases, and increases as the speed decreases, so the time required is inversely proportional to the speed of the car. Ratios A ratio is a comparison of two quantities in a particular order. Example: If there are 14 computers in a lab, and the class has 20 students, there is a student to computer ratio of 20 to 14, commonly written as 20:14. Ratios are normally reduced to their smallest whole number representation, so 20:14 would be reduced to 10:7 by dividing both sides by 2. Constant of Proportionality When two quantities have a proportional relationship, there exists a constant of proportionality between the quantities; the product of this constant and one of the quantities is equal to the other quantity. For example, if one lemon costs $0.25, two lemons cost $0.50, and three lemons cost $0.75, there is a proportional relationship between the total cost of lemons and the number of lemons purchased. The constant of proportionality is the unit price, namely $0.25/lemon. Notice that the total price of lemons, t, can be found by multiplying the unit price of lemons, p, and the number of lemons, n: . Work/Unit Rate Unit rate expresses a quantity of one thing in terms of one unit of another. For example, if you travel 30 miles every two hours, a unit rate expresses this comparison in terms of one hour: in one hour you travel 15 miles, so your unit rate is 15 miles per hour. Other examples are how much one ounce of food costs (price per ounce) or figuring out how much one egg costs out of the dozen (price per 1 egg, instead of price per 12 eggs). The denominator of a unit rate is always 1. Unit rates are used to compare different situations to solve problems. For example, to make sure you get the best deal when deciding which kind of soda to buy, you can find the unit rate of each. If soda #1 costs $1.50 for a 1-liter bottle, and soda #2 costs $2.75 for a 2-liter bottle, it would be a better deal to buy soda #2, because its unit rate is only $1.375 per 1-liter, which is cheaper than soda #1. Unit rates can also help determine the length of time a given event will take. For example, if you can paint 2 rooms in 4.5 hours, you can determine how long it will take you to paint 5 rooms by solving for the unit rate per room and then multiplying that by 5. Slope On a graph with two points, ; where and m stands for slope. If the value of the slope is positive, the line has an upward direction from left to right. If the value of the slope is negative, the line has a downward direction from left to right. Consider the following example: A new book goes on sale in bookstores and online stores. In the first month, 5,000 copies of the book are sold. Over time, the book continues to grow in popularity. The data for the number of copies sold is in the table below. # of Months on Sale - # of Copies Sold (In Thousands) 1 - 5 2 - 10 3 - 15 4 - 20 5 - 25 So, the number of copies that are sold and the time that the book is on sale is a proportional relationship. In this example, an equation can be used to show the data: , where x is the number of months that the book is on sale. Also, y is the number of copies sold. So, the slope of the corresponding line is . Finding an Unknown in Equivalent Expressions It is often necessary to apply information given about a rate or proportion to a new scenario. For example, if you know that Jedha can run a marathon (26 miles) in 3 hours, how long would it take her to run 10 miles at the same pace? Start by setting up equivalent expressions: Now, cross multiply and, solve for : So, at this pace, Jedha could run 10 miles in about 1.15 hours or about 1 hour and 9 minutes. Multiply Fractions P1. Solve the following for . (a) (b) (c) P2. At a school, for every 20 female students there are 15 male students. This same student ratio happens to exist at another school. If there are 100 female students at the second school, how many male students are there? P3. In a hospital emergency room, there are 4 nurses for every 12 patients. What is the ratio of nurses to patients? If the nurse-to-patient ratio remains constant, how many nurses must be present to care for 24 patients? P4. In a bank, the banker-to-customer ratio is 1:2. If seven bankers are on duty, how many customers are currently in the bank? P5. Janice made $40 during the first 5 hours she spent babysitting. She will continue to earn money at this rate until she finishes babysitting in 3 more hours. Find how much money Janice earns per hour and the total she earned babysitting. P6. The McDonalds are taking a family road trip, driving 300 miles to their cabin. It took them 2 hours to drive the first 120 miles. They will drive at the same speed all the way to their cabin. Find the speed at which the McDonalds are driving and how much longer it will take them to get to their cabin. P7. It takes Andy 10 minutes to read 6 pages of his book. He has already read 150 pages in his book that is 210 pages long. Find how long it takes Andy to read 1 page and also find how long it will take him to finish his book if he continues to read at the same speed. P1. First, cross multiply; then, solve for x: (a) (b) (c) P2. One way to find the number of male students is to set up and solve a proportion. Represent the unknown number of male students as the variable x: Cross multiply and then solve for x: P3. The ratio of nurses to patients can be written as 4 to 12, 4:12, or . Because four and twelve have a common factor of four, the ratio should be reduced to 1:3, which means that there is one nurse present for every three patients. If this ratio remains constant, there must be eight nurses present to care for 24 patients. P4. Use proportional reasoning or set up a proportion to solve. Because there are twice as many customers as bankers, there must be fourteen customers when seven bankers are on duty. Setting up and solving a proportion gives the same result: of patients as the variable x: . To solve for x, cross multiply: , so . P5. Janice earns $8 per hour. This can be found by taking her initial amount earned, $40, and dividing it by the number of hours worked, 5. Since , Janice makes $8 in one hour. This can also be found by finding the unit rate, money earned per hour: . Since cross multiplying yields , and division by 5 shows that , Janice earns $8 per hour. Janice will earn $64 babysitting in her 8 total hours (adding the first 5 hours to the remaining 3 gives the 8-hour total). Since Janice earns $8 per hour and she worked 8 hours, . This can also be found by setting up a proportion comparing money earned to babysitting hours. Since she earns $40 for 5 hours and since the rate is constant, she will earn a proportional amount in 8 hours: . Cross multiplying will yield , and division by 5 shows that . P6. The McDonalds are driving 60 miles per hour. This can be found by setting up a proportion to find the unit rate, the number of miles they drive per one hour: . Cross multiplying yields and division by 2 shows that . Since the McDonalds will drive this same speed, it will take them another 3 hours to get to their cabin. This can be found by first finding how many miles the McDonalds have left to drive, which is . The McDonalds are driving at 60 miles per hour, so a proportion can be set up to determine how many hours it will take them to drive 180 miles: . Cross multiplying yields and division by 60 shows that . This can also be found by using the formula (or ), where , and division by 60 shows that . P7. It takes Andy 10 minutes to read 6 pages, minutes, which is 1 minute and 40 seconds. Next, determine how many pages Andy has left to read, . Since it is now known that it takes him minutes to read each page, then that rate must be multiplied by however many pages he has left to read (60) to find the time he'll need: , so it will take him 100 minutes, or 1 hour and 40 minutes, to read the rest of his book.
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